Find the derivative of `y = sin e^(3x) `.

Answer

To find the derivative of the function \( y = \sin(e^{3x}) \), we will use the chain rule. Here are the steps to solve the problem:

### Step 1: Identify the outer and inner functions
In the function \( y = \sin(e^{3x}) \), we can identify:
- The outer function as \( \sin(u) \) where \( u = e^{3x} \).
- The inner function as \( u = e^{3x} \).

### Step 2: Differentiate the outer function
The derivative of \( \sin(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \cos(u) \]

### Step 3: Differentiate the inner function
Now, we need to differentiate the inner function \( u = e^{3x} \) with respect to \( x \):
\[ \frac{du}{dx} = e^{3x} \cdot \frac{d}{dx}(3x) = e^{3x} \cdot 3 = 3e^{3x} \]

### Step 4: Apply the chain rule
Using the chain rule, we can find the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \cos(u) \cdot 3e^{3x} \]

### Step 5: Substitute back for \( u \)
Now, we substitute back \( u = e^{3x} \):
\[ \frac{dy}{dx} = \cos(e^{3x}) \cdot 3e^{3x} \]

### Final Answer
Thus, the derivative of \( y = \sin(e^{3x}) \) is:
\[ \frac{dy}{dx} = 3e^{3x} \cos(e^{3x}) \]

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